# Inner models with large cardinal features usually obtained by forcing

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## Abstract

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal *κ* for which 2^{ κ } = *κ* ^{+}, another for which 2^{ κ } = *κ* ^{++} and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model *W* with a strongly compact cardinal *κ*, such that \({H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}\). Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + *V* = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit *δ* of <*δ*-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

## Keywords

Forcing Large cardinals Inner models## Mathematics Subject Classification (2000)

03E45 03E55 03E40## Preview

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## References

- 1.Apter A.W.: On level by level equivalence and inequivalence between strong compactness and supercompactness. Fund. Math.
**171**(1), 77–92 (2002)MathSciNetzbMATHCrossRefGoogle Scholar - 2.Apter A.W.: Diamond, square, and level by level equivalence. Arch. Math. Log.
**44**(3), 387–395 (2005)MathSciNetzbMATHCrossRefGoogle Scholar - 3.Apter A.W.: Tallness and level by level equivalence and inequivalence. Math. Log. Q.
**56**(1), 4–12 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - 4.Apter A.W.: Level by level inequivalence beyond measurability. Arch. Math. Log.
**50**(7-8), 707–712 (2011)MathSciNetzbMATHCrossRefGoogle Scholar - 5.Apter A.W., Shelah S.: On the strong equality between supercompactness and strong compactness. Trans. Am. Math. Soc.
**349**(1), 103–128 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 6.Baumgartner J.E.: Applications of the proper forcing axiom. In: Kunen, K., Vaughan, J. (eds) Handbook of Set Theoretic Topology, pp. 913–959. North-Holland, Amsterdam (1984)Google Scholar
- 7.Brooke-Taylor A.: Large cardinals and definable well-orders on the universe. J. Symb. Log.
**74**(2), 641–654 (2009)MathSciNetzbMATHCrossRefGoogle Scholar - 8.Dobrinen N., Friedman S.D.: Internal consistency and global co-stationarity of the ground model. J. Symb. Log.
**73**(2), 512–521 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - 9.Dobrinen N., Friedman S.D.: The consistency strength of the tree property at the double successor of a measurable cardinal. Fund. Math.
**208**(2), 123–153 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - 10.Foreman M.D.: Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals. Adv. Math.
**222**(2), 565–595 (2009)MathSciNetzbMATHCrossRefGoogle Scholar - 11.Friedman S.D.: Internal consistency and the inner model hypothesis. Bull. Symb. Log.
**12**(4), 591–600 (2006)zbMATHCrossRefGoogle Scholar - 12.Friedman, S.: Aspects of HOD, supercompactness, and set theoretic geology. PhD thesis, The Graduate Center of the City University of New York (2009)Google Scholar
- 13.Gitman, V., Hamkins, J.D., Johnstone, T.A.: What is the theory ZFC without power set? (submitted for publication)Google Scholar
- 14.Gitman V.: Ramsey-like cardinals. J. Symb. Log.
**76**(2), 519–540 (2011)MathSciNetzbMATHCrossRefGoogle Scholar - 15.Gitman V., Welch P.D.: Ramsey-like cardinals II. J. Symb. Log.
**76**(2), 541–560 (2011)MathSciNetzbMATHCrossRefGoogle Scholar - 16.Hamkins J.D.: Fragile measurability. J. Symb. Log.
**59**(1), 262–282 (1994)MathSciNetzbMATHCrossRefGoogle Scholar - 17.Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Log.
**101**, 103–146 (2000)MathSciNetzbMATHCrossRefGoogle Scholar - 18.Hamkins J.D.: The ground axiom. Oberwolfach Rep.
**55**, 3160–3162 (2005)Google Scholar - 19.Hamkins, J.D., Seabold, D.: Boolean ultrapowers (in preparation)Google Scholar
- 20.Kunen K.: Some applications of iterated ultrapowers in set theory. Ann. Math. Log.
**1**, 179–227 (1970)MathSciNetzbMATHCrossRefGoogle Scholar - 21.Laver R.: Making the supercompactness of
*κ*indestructible under*κ*-directed closed forcing. Israel J. Math.**29**(4), 385–388 (1978)MathSciNetzbMATHCrossRefGoogle Scholar - 22.Lévy A., Solovay R.M.: Measurable cardinals and the continuum hypothesis. Israel J. Math.
**5**, 234–248 (1967)MathSciNetzbMATHCrossRefGoogle Scholar - 23.Magidor M.: How large is the first strongly compact cardinal? or a study on identity crises. Ann. Math. Log.
**10**(1), 33–57 (1976)MathSciNetzbMATHCrossRefGoogle Scholar - 24.Mitchell W.J.: Ramsey cardinals and constructibility. J. Symb. Log.
**44**(2), 260–266 (1979)zbMATHCrossRefGoogle Scholar - 25.Reitz, J.: The ground axiom. PhD thesis, The Graduate Center of the City University of New York (2006)Google Scholar
- 26.Reitz J.: The ground axiom. J. Symb. Log.
**72**(4), 1299–1317 (2007)MathSciNetzbMATHCrossRefGoogle Scholar - 27.Solovay R.M. et al.: Strongly compact cardinals and the GCH. In: Henkin, L. (ed) Proceedings of the Tarski Symposium, Proceedings Symposia Pure Mathematics, volume XXV, University of California, Berkeley, 1971, pp. 365–372. American Mathematical Society, Providence, Rhode Island (1974)Google Scholar
- 28.Welch P.: On unfoldable cardinals, omega cardinals, and the beginning of the inner model hierarchy. Arch. Math. Log.
**43**(4), 443–458 (2004)MathSciNetzbMATHCrossRefGoogle Scholar